Chapter 5 Junctions

5.1 Introduction (chapter 3) 5.2 Equilibrium condition Contact potential Equilibrium Fermi level Space charge at a junction 5.3 Forward bias 5.3.1

(negative) Photoresist Silicon Oxide Silicon Mask/Shield/Pattern Irradiation Metal Oxide Lift off Develop

Fermi Gas and Density of State

Particle in a Infinite Well For three-dimensional box

Electron Energy Density

Density of State ρ(E)

Properties Dependent on Density of States Experiment provide information on density of state

N(E)[1-f(E)] N(E)f(E)N(E)f(E) (a) Intrinsic N(E): Density of state f(E): Probability of occupation (Fermi-Dirac distribution function) Holes (a) Intrinsic N=  N(E)dE: Total number of states per unit volume N=  N(E)f(E)dE: Concentration of electrons in the conduction band

(c) p-type(b) n-type Holes Electrons (a) Intrinsic

This density of state equation is derived from assumption of electron in the infinite well with vacuum medium, where the E is proportional to k 2. We found that the free electron in the conduction band of semiconductor has local minimum of energy E versus wave number k. We can approximate the bottom portion of the curve as if E is still proportional to k 2 and write down the similar energy-wave number equation as to describe the behavior of the free electrons, where m n * is the equivalent electron mass, which account for the electron accommodation to medium change. If we prefer to the energy at the bottom of the conduction band as a nun-zero value of Ec instead of Ec = 0, The density of state equation can be further modified as

N c : Effective density of state at bottom of C.B. N v : Effective density of state at top of V.B. n o : Concentration of electrons in the conduction band p o : Concentration of holes in the valence band E c : Conduction band edge E v : Valence band edge E F : Fermi level E i : Fermi level for the undoped semiconductor (intrinsic)

Fermi Level and Carrier Concentration of Intrinsic Semiconductor

Example 3-5 A Si sample is doped with As atoms/cm 3. What is the equilibrium hole concentration p o at 300K? Where is E F relative to E i ?

5.1 Introduction 5.2 Equilibrium condition Contact potential Equilibrium Fermi level Space charge at a junction 5.3 Forward bias 5.3.1

Electric field

Einstein relationship (explained later) Electric field

Einstein Relationship drift diffusion At equilibrium, no net current flows in a semiconductor. J p (x) = 0 Any fluctuation which would begin a diffusion current also sets up an electric field which redistributes carriers by drift. An examination of the requirements for equilibrium indicates that the diffusion coefficient and mobility must be related.

Einstein Relationship Drift Diffusion

drift diffusion Einstein Relationship Drift and diffusion diffusion

The derivation of Poisson's equation in electrostatics follows. SI units are used and Euclidean space is assumed. Starting with Gauss' law for electricity (also part of Maxwell's equations) in a differential control volume, we have: is the divergence operator. is the electric displacement field. is the free charge density (describing charges brought from outside). Assuming the medium is linear, isotropic, and homogeneous (see polarization density), then: is the permittivity of the medium. is the electric field. By substitution and division, we have: Poisson's equation